Structure-preserving neural networks for the n-body problem

In order to understand when it is useful to build physics constraints into neural net- works, we investigate different neural network topologies to solve the N-body problem. Solving the chaotic N-body problem with high accuracy is a challenging task, requiring special numerical integrators that are able to approximate the trajectories with extreme precision. In [1] it is shown that a neural network can be a viable alternative, offering solutions many orders of magnitude faster. Specialized neural network topologies for applications in scientific computing are still rare compared to specialized neural networks for more classical machine learning applications. However, the number of specialized neural networks for Hamiltonian systems has been growing significantly during the last years [3, 5]. We analyze the performance of SympNets introduced in [5], preserving the symplectic structure of the phase space flow map, for the prediction of trajectories in N -body systems. In particular, we compare the accuracy of SympNets against standard multilayer perceptrons, both inside and outside the range of training data. We analyze our findings using a novel view on the topology of SympNets. Additionally, we also compare SympNets against classical symplectic numerical integrators. While the benefits of symplectic integrators for Hamiltonian systems are well understood, this is not the case for SympNets.